Question

The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$ is

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

The correct option is C $2$

$\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& y& z\end{array}\right|=0$
applying $R_1\to R_1-R_2;R_2\to R_2-R_3$ we get
$\left|\begin{array}{ccc}x-m& n-y& 0\\ 0& y-n& r-z\\ m& n& z\end{array}\right|=0\Rightarrow (x-m)(y-n)z+(n-y)(r-z)m-n(r-z)(x-m)=0$
dividing by $(x-m)(y-n)(z-r)$ we have
$\frac{z}{z-r}+\frac{m}{x-m}+\frac{n}{y-n}=0\Rightarrow \frac{z}{z-r}+1+\frac{m}{x-m}+1+\frac{n}{y-n}+1=2$
$\Rightarrow \frac{z}{z-r}+\frac{m}{x-m}+\frac{n}{y-n}=2=2\Rightarrow \frac{z}{z-r}-1+\frac{m}{x-m}-1+\frac{n}{y-n}-1=0-1=-1$
Now $A.M\ge G.M\Rightarrow \frac{\frac{z}{z-r}+\frac{m}{x-m}+\frac{n}{y-n}}{3}\ge {(\frac{z}{z-r}+\frac{x}{x-m}+\frac{y}{y-n}=0)}^{1/3}\Rightarrow \frac{z}{z-r}+\frac{m}{x-m}+\frac{n}{y-n}\le \frac{8}{27}$